Abstract We introduce a two-grid finite difference approximation scheme for the free Schr¨ odinger equation. This scheme is shown to converge and to posses appropriate dispersive properties as the mesh-size tends to zero. A careful analysis of the Fourier symbol shows that this occurs because the two-grid algorithm (consisting in projecting slowly oscillating data into a fine grid) acts, to some extent, as a filtering one. We show that this scheme converges also in a class of nonlinear Schr¨ odinger equations whose well-posedness analysis requires the so-called Strichartz estimates. This method provides an alternative one to the one introduced by the authors in [3] using numerical viscosity. To cite this article: Liviu I. Ignat, Enrique Zuazua, C. R. Acad. Sci. Paris, Ser. I . R´ esum´ e On introduit une m´ethode bi-maille semi-discr`ete en diff´erences finies pour l’approximation num´erique de l’´equation de Schr¨ odinger. On d´emontre la convergence L2 du sch´ema et des propri´et´es de dispersivit´e uniformes par rapport au pas du maillage. Une analyse soigneuse en Fourier du symbole du sch´ema (consistant essentiellement ` a projeter des donn´ees lentes sur un maillage fin) montre que l’algorithme bi-maille agit comme un filtre des hautes fr´equences. On montre aussi la convergence du sch´ema dans une classe d’´equations non-lin´eaires dont l’´etude dans le cas continu n´ecessite des in´egalit´es de Strichartz. Cette m´ethode donne une approche alternative a ` celle introduite par les auteurs dans [3] a ` l’aide d’un sch´ema avec viscosit´e num´erique. Pour citer cet article : Liviu I. Ignat, Enrique Zuazua, C. R. Acad. Sci. Paris, Ser. I .

Email addresses: [email protected] (Liviu I. Ignat), [email protected] (Enrique Zuazua). Preprint submitted to Elsevier Science

24 f´ evrier 2010

Version fran¸ caise abr´ eg´ ee On introduit un sch´ema bi-maille en diff´erences finies pour l’approximation num´erique de l’´equation de Schr¨ odinger. Ce travail est motiv´e par [3] o` u l’on a constat´e que le sch´ema conservatif classique semi-discret en diff´erences finies ne poss`ede par les propri´et´es de dispersivit´e uniforme (par rapport au pas du maillage), n´ecessaires pour garantir la convergence pour des ´equations non-lin´eaires. Dans [3] on avait introduit un sch´ema dissipatif, incluant un terme de viscosit´e num´erique et on avait d´emontr´e la convergence du sch´ema. La viscosit´e num´erique avait comme but de dissiper les hautes fr´equences num´eriques qui ´etaient la cause du manque de dispersivit´e. Dans cette note on introduit une m´ethode bi-maille qui, ´etant de caract`ere conservatif, joue le mˆeme rˆ ole en filtrant les hautes fr´equences. Dans le cas de l’´equation de Schr¨odinger libre la m´ethode consiste simplement a projeter les donn´ees lentes d’un maillage ´epais dans un maillage fin, le rapport entre les deux mailles ´etant convenablement choisi 1/4. Cette m´ethode est inspir´ee de celle introduite par R. Glowinski dans [2] pour l’approximation num´erique des contrˆoles fronti`ere de l’´equation des ondes, dont l’efficacit´e est maintemant bien ´etablie, la convergence ayant ´et´e demontr´ee r´ecemment dans [6]. Dans le cas lin´eaire, moyennant une analyse de Fourier, et en utilisant des arguments analogues `a ceux de la th´eorie continue ([1],[4],[5] et [8]), on montre des propri´et´es de dispersivit´e (in´egalit´es de Strichartz) uniformes par rapport au pas du maillage, la convergence L2 de la m´ethode ´etant standard (consistance + stabilit´e). Dans le cas non-lin´eaire on doit introduire une discr´etisation soigneuse de la non-linearit´e, permettant de garder ses propri´et´es de cancellation au niveau des estimations d’energie, et qui ne fournit qu’une source d’oscillations lentes, admissibles dans notre sch´ema bi-maille. Ceci ´etant fait, on d´emontre la convergence du sch´ema dans le cas non-lin´eaire.

1. Introduction Let us to consider the 1-d linear Schr¨ odinger Equation (LSE) in the whole line iut + uxx = 0, x ∈ R, t > 0;

u(0, x) = ϕ(x), x ∈ R.

(1)

Its solution is given by u(t) = S(t)ϕ, where S(t) = eit∆ is the free Schr¨odinger operator which defines a unitary transformation group in L2 (R). The conservation of the L2 -norm kukL2 (R) = kϕkL2 (R) , together 0 with the classical estimate |u(t, x)| ≤ (4π|t|)−1/2 kϕkL1 (R) , leads by interpolation to the following Lp − Lp 0 result: kS(t)ϕkLp (R) ≤ t−1/2(1/p −1/p) kϕkLp0 (R) , for all p ≥ 2 and t 6= 0. More refined space-time estimates known as the Strichartz inequalities show that, in addition to the decay of the solution as t → ∞, the linear semigroup S(t) satisfies kS(·)ϕkLq (R,Lr (R)) ≤ CkϕkL2 (R) for suitable values of q and r, the so-called admissible pairs satisfying 2/q = 1/2 − 1/r. Also a local gain of 1/2 space derivate occurs in L2x,t . These properties are not only relevant for a better understanding of the dynamics of the linear system but also to derive well-posedness results for nonlinear Schr¨odinger equations ([1],[8]). In this note we introduce a two-grid finite-difference semi-discretization scheme that reproduces these properties, uniformly with respect to the mesh-size. Let us first consider the finite-difference conservative numerical scheme i

duh + ∆h uh = 0, t ∈ R; dt

uh (0) = ϕh .

(2)

Here uh stands for the infinite vector unknown {uhj }j∈Z , uj (t) being the approximation of the solution at the node xj = jh, and ∆h the classical second order finite difference approximation of ∂x2 : (∆h u)j = 2

(uj+1 − 2uj + uj−1 )/h2 . This scheme satisfies the classical properties of consistency and stability which imply the L2 convergence. The same convergence results hold for semilinear equations iuh + uxx = f (u)(NSE) provided that the nonlinearity f is globally Lipshitz. But, as proved in [8], the NSE is also well-posed for some nonlinearities that grow superlinearly at infinity. We refer to [1] for a survey of the important progresses done in this field. This well-posedness result may not be proved simply as a consequence of the L2 conservation property. Indeed, the dispersive properties of the LSE play a crucial role. Accordingly, one may not expect to prove convergence of the numerical scheme in this class of nonlinearities without similar dispersive estimates, that should be uniform on the mesh-size parameter h → 0. As we proved in [3] this conservative scheme (2) fails to have uniform dispersive properties. This is due to the high frequency spurious numerical solutions the scheme (2) introduces. We remark that there are slight but important differences between the symbols of the operators −∆ and −∆h : p(ξ) = ξ 2 , ξ ∈ R for −∆ and ph (ξ) = 4/h2 sin2 (ξh/2), ξ ∈ [π/h, π/h] for −∆h . The symbol ph (ξ) changes convexity at the points ξ = ±π/2h and has critical points also at ξ = ±π/h, two properties that the continuous symbol does not fulfill because of its strict convexity. To compensate this lack of dispersivity we propose a two-grid algorithm (inspired in [2]) and that, to some extent, acts as a filter for those unwanted high frequency components. This method is a natural alternative to the one introduced in [3], based on the use of numerical viscosity. The method is roughly as follows. We consider two meshes: the coarse one of size 4h, h > 0, 4hZ, and the fine one, hZ, of size h > 0. The method relies basically on solving the finite-difference semidiscretization (2) on the fine mesh hZ, but only for slow data, interpolated from the coarse grid 4hZ. As we shall see, the 1/4 ratio between the two meshes is important to guarantee the convergence of the method. This particular structure of the data cancels the two pathologies of the discrete symbol mentioned above. Indeed, a careful Fourier analysis of those initial data shows that their discrete Fourier transform vanishes quadratically at the points ξ = ±π/2h and ξ = ±π/h. As we shall see, this suffices to recover the dispersive properties of the continuous model.

2. Fourier Analysis of Slowly Oscillating Sequences In this section we obtain explicit properties of the discrete Fourier transform of slowly oscillating sequences (SOS). The SOS on the fine grid hZ are those which are obtained from the coarse grid 4hZ by an interpolation process. Obviously there is a one to one correspondence between the coarse grid hZ sequences and the space ChZ : supp ψ ⊂ 4hZ}. With this purpose we introduce the extension 4 = {ψ ∈ C operator E: (Eψ)((4j + r)h) =

4−r 4 ψ(4jh)

+ 4r ψ((4j + 4)h), ∀j ∈ Z, r = 0, 3, ψ ∈ ChZ 4

(3)

where δ is the Kronecker’s symbol. We now define the space V4h of slowly oscillating sequences as V4h = {Eψ : ψ ∈ C4hZ }. We also consider the projection operator Π : ChZ → ChZ 4 by (Πφ)((4j + r)h) = φ((4j + r)h)δ4r , ∀j ∈ Z, r = 0, 3, φ ∈ ChZ .

(4)

, EΠ = We remark that E : C4hZ → V4h and Π : V4h → ChZ 4 are bijective linear maps satisfying ΠE = IChZ 4 hZ h e IV4h , where IX denotes the identity operator on X. We now define Π = EΠ : C → V4 , which acts as a smoothing operator and associates to each sequence on the fine grid a slowly oscillating sequence. As we said above the restriction of this operator to V4h is the identity. Concerning the discrete Fourier transform of a SOS by means of explicit computations one can prove that: 3

Lemma 2.1 Let φ ∈ l2 (hZ). Then c ec Πφ(ξ) = 4 cos2 (ξh) cos2 (ξh/2)Πφ(ξ).

(5)

ec Remark 1 A simpler construction may be done interpolating 2hZ sequences. We then get Πϕ(ξ) = 2 c 2 cos (ξh/2)Πϕ(ξ). This cancels the spurious numerical solutions at the frequencies ±π/h, but not at ±π/2h. In this case, as we proved in [3], the Strichartz estimates fail to be uniform on h. Thus we rather choose the ratio between grids to be 1/4.

3. Estimates of the Linear Semigroup As we proved in [3], there is no gain (uniformly in h) of integrability of the linear semigroup eit∆h . However, there are subspaces of ChZ , namely V4h , where the linear semigroup has appropriate decay properties, uniformly on h > 0. The main results we get are the following Theorem 3.1 Let p ≥ 2. The following properties hold : 0

0

p e lp (hZ) . |t|−1/2(1/p −1/p) kΠϕk e i) keit∆h Πϕk lp0 (hZ) for all ϕ ∈ l (hZ), h > 0 and t 6= 0.

e belongs to Lq (R, lr (hZ)) ∩ C(R, l2 (hZ)) ii) For every sequence ϕ ∈ l2 (hZ), the function t → eit∆h Πϕ it∆h e e l2 (hZ) uniformly on h > 0. for every admissible pair (q, r). Furthermore ke ΠϕkLq (R,lr (hZ)) . kΠϕk R e (s)dskLq (R,lr (hZ)) . kΠF e kLq˜(R,lr˜(hZ)) iii) Let (q, r), (˜ q , r˜) be two admissible pairs. Then k s

R

kT (·)f kLq (R,lr (Z)) . kf kl2 (Z) and T (t − s)F (s)ds . kF kLq˜(R,lr˜(Z)) . (6) s

Lq (R,lr (Z))

Finally we use the definition of T (t) in order to obtain the estimates for eit∆1 . Concerning the local smoothing properties we can prove that Theorem 3.2 Let r ∈ (1, 2]. Then 2 R ∞ e )j dt . kΠf e k2r supj∈Z −∞ (D1−1/r eit∆h Πf l (hZ) for all f ∈ lr (hZ), uniformly in h > 0. 4

(7)

e = T (t)Πf it is sufficient Sketch of the Proof. By scaling we can assume that h = 1. Using that eit∆1 Πf 2 R ∞ 1−1/r 2 to prove that supj∈Z −∞ (D T (t)ψ)j dt . kψklr (Z) . We now introduce the continuous extension of T: Rπ 2 ξ ixξ (T1 (t)ϕ)(x) = −π e−4it sin 2 ϕ(ξ)e ˆ cos2 ξ cos2 (ξ/2)dξ. (8) It is sufficient to prove that 2 R∞ supx∈R −∞ (D1−1/r T1 (t)ϕ)(x) dt . kϕk2Lr (R)

(9)

for all ϕ ∈ Lr (R) with supp ϕˆ ∈ [−π, π]. Using Sobolev’s imbedding Lr (R) ,→ H 1/2−1/r (R) the inequality (9) may be reduced to the following one R∞ 2 supx∈R −∞ |(T1 (t)ϕ)(x)| dt . kD−1/2 ϕk2L2 (R) for all ϕ ∈ S(R). (10) Applying the results of [5] (Theorem 4.1, p. 54) to T1 we get R∞ R π ˆ 2 4 ξ cos4 (ξ/2) Rπ 2 supx∈R −∞ |(T1 (t)ϕ)(x)| dt . −π |f (ξ)| cos dξ . −π | sin ξ|

|fˆ(ξ)|2 |ξ| dξ

. kD−1/2 f k2L2 (R) .

(11)

4. A conservative approximation of the NSE We concentrate on the semilinear NSE equation in R with repulsive power law nonlinearity : iut + ∆u = |u|p u, x ∈ R, t ∈ R;

u(0, x) = ϕ(x), x ∈ R.

(12)

As proved in [8], (12) is globally well posed for all ϕ ∈ L2 (R) and p ∈ [0, 4). We consider the following semi-discretization i

duh e (uh ), t ∈ R; + ∆h uh = Πf dt

e h, uh (0) = Πϕ

(13)

where f (uh ) is a suitable approximation of |u|p u with 0 < p < 4. In order to prove the global well-posedness of (13), we need to guarantee the conservation of the l2 (hZ) norm of solutions, a property that the solutions e (uh ), uh )l2 (hZ) ∈ R. The of NSE satisfy. For that the nonlinear term f (uh ) has to be chosen such that (Πf following holds: e (u)k (p+2)0 Theorem 4.1 Let be p ∈ (0, 4), q = 4(p + 2)/p and f : ChZ → ChZ satisfying kΠf l (hZ) . p h 2 e k|u| ukl(p+2)0 (hZ) and (Πf (u), u)l2 (hZ) ∈ R. Then for every ϕ ∈ l (hZ), there exists a unique global solution uh ∈ C(R, l2 (hZ)) ∩ Lqloc (R; lp+2 (hZ))

(14)

of (13) which satisfies the following estimates e l2 (hZ) and kuh kLq (I,lp+2 (hZ)) ≤ c(I)kΠϕk e l2 (hZ) kuh kL∞ (R,l2 (hZ)) ≤ kΠϕk

(15)

for all finite interval I, where the above constants are independent of h. Remark 2 The conditions above on the nonlinearity satisfied if one choose P3 h h (u + u )) 4 ; g(s) = |s|p s. (f (uh ))4j = g (uh4j + r=1 4−r 4j+r 4j−r 4

(16)

h p h e (uh )k (p+2)0 With this choice it is easy to check that kΠf l (hZ) ≤ Ck|u | u kl(p+2)0 (hZ) with C > 0 indepene (uh ), uh )l2 (hZ) ∈ R as follows dently of h > 0. Furthermore (Πf

5

h r 4−r h h e (uh ) , uh )l2 (hZ) = h P3 P (Πf r=0 j∈Z 4 (f (u ))4j + 4 (f (u ))4j+4 u4j+r P P3 r h P 3 4−r h u + u = h j∈Z (f (uh ))4j 4j+r r=0 4 4j+r−4 r=0 4 P P P3 3 h h h h ))/4 (uh4j + r=1 4−r = h j∈Z g (uh4j + r=1 4−r + u (u 4j−r 4j+r 4 4 (u4j+r + u4j−r )). Sketch of the Proof. Local existence and uniqueness are consequence of the Strichartz estimates (Theoh h e rem 3.1) and a fixed point P argument. The fact that (Πf (u ), u )l2 (hZ) is real guarantees the conservation of the discrete energy h j∈Z |uj (t)|2 . This allows excluding finite-time blow-up. In the sequel we consider the piecewise constant interpolator Ih . We choose (ϕhj )j∈Z , an approximation e h * ϕ weakly in L2 (R). of the initial data ϕ ∈ L2 (R), such that Ih Πϕ The main convergence result is the following Theorem 4.2 Let uh be the unique solution of (13). Then the sequence Ih uh satisfies ?

Ih uh * u in L∞ (R, L2 (R)), h

Ih u * u in

(17)

Lqloc (R, Lp+2 (R)),

(18)

Ih uh → u in L2loc (R × R),

(19)

0

e (uh ) * |u|p u in Lq (R, L(p+2)0 (R)) Ih Πf loc

(20)

where u is the unique solution of NSE and 2/q = 1/2 − 1/(p + 2). Sketch of the Proof. Using the result of Theorem 3.2 with r = 2 for initial data and r = (p + 2)0 for e h kl2 (Z) . Then, by a compactness argument nonlinearity we first prove that kIh uh kL2 (R,H 1/(p+2) (R)) ≤ kΠϕ h loc

loc

(see [7]), we can extract a subsequence converging locally strongly in L1x,t . This, together with the uniform (with respect to h) estimates of Theorem 4.1, suffices to obtain the stated convergence results. In particular, it suffices to pass to the limit in the nonlinear term and to identify the limit as the solution of NSE. Remark 3 Our method works similarly in the critical case p = 4 for small initial data. Remark 4 The techniques and results of this paper extend to the LSE and NSE in several space dimensions.

Acknowledgements The authors thank Fernando Soria for fruitful discussions. This work has been supported by Grant BFM2002-03345 of the Spanish MCYT and he Network “New materials, adaptive systems and their nonlinearities: modelling, control and numerical simulation” of the EU. Liviu I. Ignat was supported by a doctoral fellowship of MEC (Spain). References [1] T. Cazenave. Semilinear Schr¨ odinger Equations. Courant Lecture Notes, 10, 2003. [2] R. Glowinski. Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation. J. Comput. Phys., (2), 103, 189–221, 1992. [3] L. I. Ignat and E. Zuazua. Dispersive properties of a viscous numerical scheme for the schr¨ odinger equation. C. R. Acad. Sci. Paris, Ser. I, 2005 (to apper).

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[4] M. Keel and T. Tao. Endpoint Strichartz estimates. Am. J. Math., (5), 120, 955–980, 1998. [5] C. E. Kenig, G. Ponce, and L. Vega. Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J., (1), 40, 33–69, 1991. [6] M. Negreanu and E. Zuazua. Convergence of a multigrid method for the controllability of a 1-d wave equation. C. R. Math. Acad. Sci. Paris, Ser. I, (5), 338, 413–418, 2004. [7] J. Simon, Compact sets in the space Lp (0, T ; B). Ann. Mat. Pura Appl., (4), 146, 65-96, 1987. [8] Y. Tsutsumi, L2 -solutions for nonlinear Schr¨ odinger equations and nonlinear groups. Funkc. Ekvacioj, Ser. Int., 30, 115–125, 1987.

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